Effect of Fiber Geometry and Representative Volume Element on Elastic and Thermal Properties of Unidirectional Fiber-Reinforced Composites

The aim of present work is focused on the evaluation of elastic and thermal properties of unidirectional fiber-reinforced polymer composites with different volume fractions of fiber up to 0.7 usingmicromechanical approach. Twoways for calculating thematerial properties, that is, analytical and numerical approaches, were presented. In numerical approach, finite element analysis was used to evaluate the elastic modulus and thermal conductivity of composite from the constituent material properties. The finite element model based on three-dimensional micromechanical representative volume element (RVE) with a square and hexagonal packing geometrywas implemented by using finite element codeANSYS. Circular cross section of fiber and square cross section of fiberwere considered to develop RVE. The periodic boundary conditions are applied to the RVE to calculate elastic modulus of composite. The steady state heat transfer simulations were performed in thermal analysis to calculate thermal conductivity of composite. In analytical approach, the elastic modulus is calculated by rule of mixture, Halpin-Tsai model, and periodic microstructure.Thermal conductivity is calculated analytically by using rule of mixture, the Chawla model, and the Hashin model. The material properties obtained using finite element techniques were compared with different analytical methods and good agreement was achieved. The results are affected by a number of parameters such as volume fraction of the fibers, geometry of fiber, and RVE.


Introduction
There has been a considerable increase in the use of fiber composite materials in various industries like aerospace, automotive, infrastructures, and sporting goods due to their specific properties like strength, stiffness, toughness, high corrosion resistance, high wear resistance, high chemical resistance, and reduced cost.These materials can take advantage of different properties of their constituents, microstructure, and interaction between constituents in order to improve the mechanical behavior of parts made from them.The mechanics of fiber-reinforced composites are complex due to their anisotropic and heterogeneous characteristics.The evaluation of effective mechanical and thermal properties of composite materials is of paramount importance in engineering design and application.Generally, two approaches are considered in obtaining the global properties of composites: (a) macromechanical analysis and (b) micromechanical analysis.In macromechanical analysis the composite material is considered as a homogeneous orthotropic continuum.In micromechanical analysis the study of composite material is at the fiber and matrix level.Typically the unit cell technique combined with the known material properties of fiber and matrix is used to determine the overall behavior of the composite [1].A number of methods have been developed to predict and to simulate the mechanical and thermal behavior of composites.Basic analytical approaches have been reported [2][3][4] to predict the composite materials properties, for example, strength, stiffness, and thermal conductivity.Prediction of boron and aluminium composite properties from a representative volume element (RVE) with square and hexagonal geometry has been reported [5].Micromechanical analysis of unidirectional fiber-reinforced composites with square and hexagonal unit cells has been reported [6,7] to evaluate the effective material properties.
Patnaik et al. [8] studied the micromechanical and thermal characteristics of glass-fiber-reinforced polymer composites.The experimental results were in good agreement with finite element model based on representative area element approach.Melro et al. [9] predict the inelastic deformation and fracture of randomly distributed unidirectional fiber-reinforced polymer composite materials with different RVEs.Several researchers made their significant contribution in studying thermal characteristics of fiber composites.Springer and Tsai [10] studied the composite thermal conductivities of unidirectional composites and expressions are obtained for predicting these conductivities in the directions along the filaments and normal to them.Islam and Pramila [11] predict the effective transverse thermal conductivity of fiber-reinforced composites by using finite element method.Square and circular cross section fibers were used for perfect bonding at fiber-matrix interface and with interfacial barrier by using four different sets of thermal boundary conditions.Al-Sulaiman et al. [12] predict the thermal conductivity of the constituents of fiber-reinforced composite laminates using three empirical formulas.Grove [13] computed transverse thermal conductivity in continuous unidirectional fiber composite materials using finite element and spatial statistical techniques for a range of fiber volume fractions up to 0.5.Lu [14] used boundary collocation scheme for calculation of transverse effective thermal conductivity of 2-dimensional periodic arrays of long circular and square cylinders with square array and long circular cylinders with hexagonal array for a complete range of fiber volume fractions.
Although a great deal of work has already been done on fiber-reinforced polymer composites with circular cross-section of fiber, square cross section of fiber with different RVE models using finite element analysis is hardly been reported.To this end, the objective of the present work is developing a three-dimensional micromechanical RVE with a square and hexagonal packing geometry with circular and square fiber cross sections.A numerical homogenization technique based on the finite element analysis was used to evaluate the elastic modulus and thermal conductivity of composite.The finite element results are compared with the analytical methods.The aim is to demonstrate applicability of homogenization technique by using finite element method to predict material characteristics in advance.

Materials and Methods
In this present investigation, unidirectional glass fiber as reinforcement phase and epoxy as matrix phase for the composite material were considered.The fiber and matrix materials are considered as isotropic and homogeneous.The properties of the constituent materials are as shown in Table 1.In a real unidirectional fiber-reinforced composite, the fibers are arranged randomly and it is difficult to model random fiber arrangement.
For this analysis, circular and square cross section fiber composite material is considered.The schematic diagram of the unidirectional fiber composite where the fibers are arranged in the square and hexagonal array is shown in Figure 1.By varying the volume fraction of fiber from 0.1 to 0.7 the elastic and thermal properties of composite material are determined.

Constitutive Equations for Fiber Composite Material.
The most general form of the anisotropic constitutive equations for homogeneous and elastic composite materials is given by Hook's law as shown in (1) [15].Consider where   and   are normal and shear components of stress, respectively,   and   are the normal and shear components of strain, respectively, and   is the symmetric stiffness matrix with 21 independent, elastic constants.According to their behaviour, composites may be characterized as generally anisotropic, monoclinic, orthotropic, and transversely isotropic.In present work, transversely isotropic characteristics have been considered for the fiber-reinforced composite.
A transversely isotropic material is to be a material whose

Glass fiber Matrix
Square unit cell effective properties are isotropic in one of its planes and the stiffness tensor is represented in Once the components of the transversely isotropic stiffness tensor  are known, the elastic properties of homogenized material can be computed by (3) [16].Consider where  1 ,  2 , ] 12 , and  23 are longitudinal modulus, transverse modulus in plane Poisson's ratio, and in plane shear modulus, respectively.

Generation of RVE.
For simplicity reasons, most micromechanical models assume a periodic arrangement of fibers for which a RVE or unit cell can be isolated.The RVE has the same elastic constants and fiber volume fraction as the composite.The periodic fiber sequences commonly used are the square array and the hexagonal array.For a square packing RVE as shown in Figures 2(a) and 2(c) the maximum theoretically achievable fiber volume fraction is 78.54%.For square RVE the diameter of fiber is calculated by where   is volume fraction of fiber;  1 ,  2 , and  3 are the length of square RVE; and   is the diameter of fiber.For the hexagonal packing RVE as shown in Figures 2(b) and 2(d) the maximum theoretically achievable fiber volume fraction is 90.69%.Obviously, with a hexagonal packing geometry a composite can be made more compact than with a square packing geometry.For hexagonal RVE the diameter of fiber is calculated by where  3 =  2 tan(60 ∘ ) and  2 = 4 1 .

Finite Element Modeling
In order to evaluate the effective properties of composite, the finite element software package ANSYS is used.The program  is written in APDL (ANSYS Programming Design Language), which is delivered by the software and it makes the handling much more comfortable.For simplification, there are many assumptions considered for the present analysis such as fibers which are arranged in a particular pattern (square and hexagonal) in a matrix.The composite is free of voids and other irregularities, all fibers are uniformly distributed in the matrix and perfectly aligned, and the interface between the fiber and matrix is perfectly debonded.In the study of the micromechanics of fiber-reinforced materials, it is convenient to use an orthogonal coordinate system that has one axis aligned with the fiber direction.The axis 1 is aligned with the fiber direction, the axis 2 is in the plane of the RVE and is perpendicular to the fibers, and the axis 3 is perpendicular to the plane of the RVE and is also perpendicular to the fibers as shown in Figure 2. Dimensions considered for the analysis are  1 = 1.0 × 10 −5 m,  2 = 1.0 × 10 −5 m, and  3 = 1.0 × 10 −5 m for square RVE.For hexagonal RVE  1 = 1.0 × 10 −5 m,  2 and  3 are calculated by using (5).The radius of fibers corresponds to volume fractions ranging from 0.1 to 0.7.Three-dimensional structural solid element SOLID186 is used to determine elastic properties and is defined by 20 nodes having three degrees of freedom at each node.They are translations in the nodal 1, 2, and 3 directions.For thermal conductivity a threedimensional quadratic brick element SOLID90 is used for discretization of the constituents and is defined by 20 nodes with a single degree of freedom (temperature) at each node.The meshed model of square and hexagonal RVE at 0.4 of fiber volume fraction is shown in Figure 3.

Boundary Conditions for Evaluation of Elastic Properties.
Composite materials can be represented as a periodic array of the RVEs.Therefore, the periodic boundary conditions must be applied to the RVE models.This implies that each RVE in the composite has the same deformation mode and there is no separation or overlap between the neighboring RVEs after deformation [17,18].The resumed boundary conditions applied are given in Table 2.Note that , , and  are the displacements along 1, 2, and 3 directions, respectively, applied on the AEDH, BFCG, ABCD, EFGH, DHGC, and AEFB faces as shown in Figure 2.After applying boundary conditions and the displacement constant, the corresponding  engineering constants are calculated as follows in terms of corresponding stresses and strains shown in (6): where  and  are the average stresses and average strains and  is the volume of the RVE.The elastic properties can be calculated by using the constitutive equations of the material properties as the ratio of corresponding average stresses and average strains as shown in (3).Figures 4 and 5 show the counter of stress and strain in square and hexagonal RVE at 0.4 of volume fraction.

Boundary Conditions for Evaluation of Thermal Conductivity.
The steady state heat transfer simulations are performed by using finite element analysis to predict thermal conductivity of composite along the longitudinal and transverse direction.The thermal boundary conditions considered in the present analysis are shown in Figure 2  the transverse modulus increases with increase in fiber volume fraction.The transverse modulus evaluated by finite element analysis with hexagonal RVE is more close to the Halpin-Tsai model and periodic microstructure as compared to the results obtained from rule of mixture and finite element analysis with square RVE.
Figure 10 shows the effect of fiber volume fraction on the in-plane Poisson's ratio of composite.It is evident from the figure that the major Poisson's ratio decreases with increase in the volume fraction of fiber due to increase in material resistance.The finite element results are in good agreement with analytical methods.
In-plane shear modulus of composite is the ratio of shear stress to the shear strain in longitudinal direction.shows the effect of fiber volume fraction on the in-plane shear modulus of composite.It is clear from the figure that the shear modulus increases with increases in fiber volume fraction.Also, it can be observed that there is a good agreement between results obtained from finite element analysis and hexagonal RVE with Halpin-Tsai model and periodic microstructure as compared to rule of mixture and finite element analysis with square RVE.

Effect of Volume
Fraction on Thermal Conductivity.Longitudinal thermal conductivity of composite is the property of a material to conduct heat in parallel to the direction of the fibers.Figure 12 shows the effect of fiber content on the longitudinal thermal conductivity using rule of mixture,

Figure 1 :
Figure 1: Arrangement of fibers in (a) square array with circular fibers, (b) hexagonal array with circular fibers, (c) square array with square fibers, and (d) hexagonal array with square fibers.

Figure 3 :Figure 4 :
Figure 3: Meshed model of (a) square RVE with circular fiber, (b) hexagonal RVE with circular fiber, (c) square RVE with square fiber, and (d) hexagonal RVE with square fiber.

Figure 9 :
Figure 9: Transverse modulus validation with different volume fraction of fiber.

Figure 10 :Figure 11 :
Figure 10: In-plane Poisson's ratio validation with different volume fraction of fiber.

Table 2 :
Boundary conditions along the 1, 2, and 3 directions of the RVE.